QUANTUM INTERACTION 2013, LEICESTER, UK

A Quantum Framework for ‘Sour grapes’ in Cognitive dissonance.

Polina Khrennikova,

School of Management,

University of Leicester

COGNITIVE DISSONANCE BIAS

• The bias of cognitive dissonance firstly discovered by Leon Festinger in 1950th

• Is often referred to as an action- opinion theory, where actions can influence a person’s opinions , beliefs, identity, see also Self- Perception theory by Bem.

• The term refers to various situations ( contexts) were the individual is faced with conflicting cognitions ( e.g a smoker, a person engaging in an unpleasant activity to reach a goal)

• An tension in the person’s mental state occurs influenced by his/ her beliefs, emotions, attitudes, identity, and other dissonant elements

• The choices, opinions and actions are influenced by the personal endeavor to balance the conflicting cognitions and restore the mental harmony

• ‘The agents that are comfortable with dissonance will likely be able to maintain attitudes that do not conform to their actions while those who prefer a consistent cognitive state will experience a significant swing in attitude as a result of actions that they choose to take’ (Kitto, Boschetti, Bruza, 2012, p.8)

COGNITIVE DISSONANCE AND VIOLATION OF BAYESIAN UPDATING

• Cognitive dissonance type of behavior is biased and inconsistent with the postulates of rational homo economicus.

• Incorrect updating of new information and the violation of classical probabilistic framework takes place.

• Individuals are processing information incompletely ignoring some factors not to cause uneasiness and cognitive discomfort

- making excuses and lowering the significance of the dissonant element

- or exaggerating the importance of some factors

• Disjunction and Conjunction errors take place

e.g the information about unhealthiness of smoking, liking of an ‘ unpleasant boring’ task or job.

OUTLINE OF THE PAPER

To show how this type of behavior works in practice we present :

• a) a ‘gedanken experiment’ in a simplified context

• b) illustrative experiment ‘the forbidden toy paradigm’ by Aronson and Carlsmith (1963 ) with real data.

• After presenting the experimental data we will strive for a solution of the cognitive dissonance problem with the help of the quantum framework

• We use the quantum probabilistic framework to find the interference effect and to test if Born’s rule can be applied for decision probabilities

BAYES RULE AND THE LAW OF TOTAL PROBABILITY

• Bayesian probability widely applied in modern economics and decision making through the 20th Century is explicating a particular concept of rational choice.

• The Bayes Formula for an event A ( also called prior probability) is updated as new event B takes place (information, data).

• P (A|B) depicts the conditioning of A by the new event B, also known as posterior probability.

• The Bayes formula can be also expressed thought the Law of Total Probability (right hand side)

THEORETICAL GEDANKEN EXPERIMENT

• The aim is to show in a simplified context the violation of Bayesian updating procedure occurring with Cognitive Dissonance.

• Part A: The children are asked to choose a toy which they would mostly like to play with ( we depict G+). After the experiment takes place where each child is left in a room with a variety of toys for 10 min and covertly observed. Two threat contexts are introduced: one group of the children is told that they are severely prohibited to play with the favorite toy ( S) the second group are mildly prohibited to play with the favorite toy (M).

• In line with the postulates of rationality and Bayes updating an “inequality of rationality” should hold:

• (1)

• Part B: The experimenter returns to the room and ‘removes’ the experimental context, allowing each child to play with any toy including the favorite one. The children who play with the experimental toy (G+) are asked in which groups they were: S or M.

• The probabilities of observed data (S|G+) and (M|G+) are obtained ( the likelihood function)

“GEDANKEN EXPERIMENT” – OUTCOMES

Bayes formula would give :

(2) (3)

P (G+) for both contexts = 1

P (S) = P(M)= ½

We can *switch* the probabilities and (1) would predict that following “contextual behavior inequality” would hold:

(4)

• Cognitive dissonance context gives a violation of (4) and consequently (1) involving a mismatching of the Bayesian updating procedure.

• This gives an indication of the impossibility to use the apparatus of classical probability theory for such decision making context.

EXPERIMENT FORBIDDEN TOY PARADIGM

• Cognitive dissonance is measured indirectly via the change of a toy’s attractiveness among the participants.

• The aim of the experiment is to show that cognitive dissonance exists and can be enhanced or reduced in this example by the level of prohibition.

• Two threat contexts Mild and Severe are applied. The design is similar to our ‘Gedanken experiment’.

• Here the context is multipart and instead the desirability of the toy before and after the experimental conditions is measured. It is established through a ranking of the toys before and after the experiment.

• We denote it by L+/ L- (the same or increased attractiveness of the toy or decreased attractiveness of the toy)

• Note: In both threat contexts none of the children plays with the toy

METHOD

• 22 preschool children ( 11 boys and 11 girls ranging in age from 3.8 to 4.6 years)

• all children took part in both experimental conditions (Mild and Severe) with a time interval of 45 days

• Part A: a ranking of the toys two by two, until a choice between 10 pairs of toys is established. The experimenter takes second ranked toy places it on a low board and leaves the room. The child is observed for 10 min through a one way mirror .

• Part B: the experimenter comes back and gives each child a chance to play with the toys again. After the second ranking list is established.

• an increase of attractiveness of the toy after the threat in the Severe condition and a decrease in the Mild condition is observed.

• the authors make an additional experiment on 11 children to establish a baseline for the effect of ´increased desirability’ in S context

ANALYSIS OF EXPERIMNTAL DATA

• The results support the hypothesis that cognitive dissonance arises in M context ( the children reduce the dissonance by lowering the attractiveness of the key toy)

• The data from the experiment

gives following frequencies:

• We check if the Formula of total probability holds:

The principle of additivity of FTP is violated

• Similar violation of FTP as observed in the disjunction effect in Savage Sure Thing Principle violation

• we cannot directly envisage whether the Bayes formula was violated because of the altered context

P(L+|M) 0.636

P(L-|M) 0.364

P(L+|S) 1

P(L-|S) 0

MOTIVATION FOR USAGE OF QUANTUM FRAMEWORK

• The aim of applying the QL representation of observables is to predict the prior probability with the aid of conditional probabilities and

• From quantum physics we borrow the notion of incompatible observational contexts that lead to complementarity of quantum measurements

• The usage of quantum probabilistic framework provides us a numerical measure of incompatibility of cognitions (the so-called coefficient of interference)

• In our case: cognitive dissonance phenomenon is regarded as interference of children’s cognitions that are incompatible.

• the emotional part also plays an important role namely the liking for the toy. It can also be incompatible with the cognition part of the mental state ( see eg. Busemeyer and Bruza, 2012)

• Mathematically : probability amplitudes for L+ and M or S threat conditions ( to ‘like the toy’ and ‘not to be allowed to play with it’) interfere with each other

QUANTUM PROBABILISTIC REPRESENTATION OF DATA

we apply the quantum probability formula as an extension of the traditional probability equation with the ‘interference term’.

 

For our problem it has the form:

we obtain:

angle θ= 1.34 radian

cos θ= 0.228 – Positive interference of probability amplitudes for our observables

QUANTUM PROBABILISTIC REPRESENTATION OF DATA

• interference of probability amplitudes for severe punishment condition and the liking of the toy possibly give positive interference?

• Born’s rule check : whether we can use the final probability amplitudes to determine the prior probability ( L+ or L-)

•

• = 0.52+ 0.475 = 0.995≈ 1

• Born’s Rule holds and we can proceed by using it to reconstruct the wave function from transition probabilities and decision making context.

QUANTUM PROBABILISTIC REPRESENTATION OF DATA

• Transition probabilities for our experiment

• we note that the matrix is not doubly stochastic as it should be in quantum physics:

• Statistics of the experiment is neither quantum nor classical. The ‘Quantumness’ is merely present in the phenomenological application of mathematical calculus.

• Observables (in our case the choices L+/L- and events S/M) are not completely captured by the two dimensional Hilbert space and a state space of higher dimension would be needed.

SUMMARIZING REMARKS

• we observed non – classicality of children's behavior, were Kolmogorov’s probabilistic framework was violated

• an illustration of a direct violation of the Bayes formula ( in the gedanken experiment part) was shown

• We applied to our problem the quantum probabilistic framework and found a positive interference of transition probability amplitudes

• We checked whether Born’s rule can be applied for this context to obtain the final probabilities

• As a next step we propose to use dynamical quantum equation for modeling the state transition and finding final choice probabilities

• we remark that there is no claim about universality of Quantum framework possibly alternative classical probabilistic frameworks could be suggested

• We primarily strive to accurate and more general mathematical framework

• thank you for your attention